Questions about alloy solidification

[bamboomoon.harold]

Hello everyone

I have a few questions about the equations in the alloysolidification application. First, I don't quite understand why there are some existing comments in the equation.cc of the alloysolidification application, like the picture below

Second, could you please provide more details about grad_phi and grad_U dot product term? I saw that the code divided by two u_con items, but I didn't understand why when I checked the documentation.

Third, the value term of xi is shown in the picture below, but when I looked at the documentation, I noticed that the code was not quite the same as the equations in the documentation, and I found that the cubic term of φ did not exist in the code, as it is in the documentation.

Thanks in advance!

[Toddy Liu]

Hello, Harold

I think I can clarify some of your questions and hope it can work.

The first problem,I think it depends on the phase field equation. Some articles used tep term to calculate u_phi while here used a_n.

The third question, if you expand the equation written in the figure you post, you may get the cubic term of phi.

[Toddy Liu]

As for the second question, I feel confused when I checked the documentation for  the first time. And I think the governing equation for U within the documentation may have something wrong. After reading some references, I found one which was very close to the equation.cc code, please refer to Echebarria B, Folch R, Karma A, et al. Quantitative phase-field model of alloy solidification[J]. Physical review E, 2004, 70(6): 061604.

According to the reference, let's correct the U equation and doing time discretization and getting weak formulation, the details are as follows,  it's a littile bit long but I think it covers all the details during the derivation. If there were mistakes, welcome your corrections.

[bamboomoon.harold]

Thank you Toddy

[bamboomoon.harold]

Hello Toddy

Thank you for the derivation, it is very helpful to me. However, there is still one thing I don't understand, which is how  is transformed into .

Thanks!

[Toddy Liu]

Hi, Harold

It is nothing but the definition of derivative. 

[bamboomoon.harold]

Thank you Toddy!

[David Montiel Taboada]

Thank you, Toddy and Harold, for your input!

I will revise the equations in the documentation of the app and make sure they are correct (and the derivation is clear). 

Regarding the equations that are commented out, I think they were used to calculate the term u_phi differently (as a function of temperature rather than U). 

I have now committed those changes in the repository. 

David

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[Paul Gabriel Lerona]

Hi all,

I know this question was posted about a year ago already but I have a few questions. 

1. In the formulation of the original paper, the coefficient before phi is [1-(1-k)...] 

However, in the equations.cc file,

It is stated as 1+(1-k). Am I missing something here? Furthermore, in the formulation of the application, it is however correct,

2. The second concern is in the formulation, I think the negative sign for the antitrapping current is distributed twice,

while in the model

however, I think it is coded correctly 

Thank you 

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[David Montiel Taboada]

Thank you, Paul

Let me look into it. 

David

[stv...@gmail.com]

Hi David,

I want to pick this thread back up because I think there are some inconsistencies between the documentation and the implementation of alloySolidification.

The Echebarria et al. paper (10.1103/PhysRevE.70.061604) that is referenced gives two ways to calculate the relaxation time, tau0. One in Eq 121 and one in Eq 123. The governing equations they give in Eqs 132-133 use the Eq 121 expression (for reasons that I don't believe are clearly stated). These are the equations in the alloySolidification documentation. However, the actual code seems to use the Eq 123 expression. That expression was used in the 2008 Farzadi et al. paper (doi:10.1088/0965-0393/16/6/065005, also for reasons that I don't believe are clearly stated). The documentation and the code need to match. I'd also be interested to know what motivated the use of the Eq 123 expression.

Second, the code has a parameter U_off that shifts the interface temperature. This variable shows up in the code but is not in the governing equations.

Third, the code has a parameter y0 that controls the radius of the initial seed. The documentation seems to imply that y0 controls the location of a planar interface.

If we can sort out what the documentation should and shouldn't say, I might be able to create a PR to fix the issues.

Best,

Steve

[David Montiel Taboada]

Hello, Steve

Thank you for catching those inconsistencies and pointing them out. 

Let me look into my notes to see if I can figure out why we are using Eq 123 rather than 121 (if there is any reason at all). I'll let you know.

I think I can fix the other two issues. 

David

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[stv...@gmail.com]

Hi David,

I was wrong about y0 -- it control the reference location for the reference temperature (i.e. at what y value T is T_0 at t=0). It has nothing to do with the radius of the seed. That also means that it is very close to what U_off does. They show up in the code like this:

U + U_off + (y-y0-Vt)/l_t

[David Montiel Taboada]

Yes, I think the idea was for y0 to define the initial position of the interface and for U_off to set the initial undercooling at the interface.

I will go over the documentation and the code with all your comments and see where it needs clarification. 

David

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[David Montiel Taboada]

Hello, everyone

I have addressed almost all of the issues brought up in this conversation. Thank you everyone for spotting them and bringing them to my attention. 

I still need to check why the cubic term in phi is missing in the code. Adding it to the equations results in very different results with non-physical interface morphologies.

Best,

David

[David Montiel Taboada]

Upon closer inspection, I realized that the cubic term appears if the last (1-phi) term below is distributed to multiply phi and the term proportional to lambda. So the cubic term is not missing. 

I will rewrite this equation in the code to make it obvious.

Best, 

David

scalarvalueType eq_xi =

((phi-constV(lamda)*(U+tep+constV(U_off))*(constV(1.0)-phi*phi))*(constV(1.0)-phi*phi));